Pierre Deligne

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Pierre René Deligne is a Belgian mathematician best known for his profound contributions to algebraic geometry, number theory, and related fields. Born in 1944, he gained international recognition for his work proving the Weil conjectures—an achievement that significantly advanced modern algebraic geometry. Deligne received the Fields Medal in 1978 for this and other groundbreaking results. He has been associated with the Institute for Advanced Study in Princeton, where his research has influenced numerous areas of mathematics, including Hodge theory, arithmetic geometry, and representation theory. Over his career, he has also been honored with several prestigious awards, such as the Crafoord Prize, the Wolf Prize, the Abel Prize, and the Shaw Prize, highlighting the lasting impact of his work.

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La conjecture de Weil. I

1974-12-01

Pierre Deligne

The irreducibility of the space of curves of given genus

1969-01-01

Pierre Deligne , D. Mumford

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Théorie de hodge, III

1974-01-01

Pierre Deligne

Equations Différentielles à Points Singuliers Réguliers

1970-01-01

Pierre Deligne

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Théorie de Hodge, II

1971-12-01

Pierre Deligne

La Conjecture de Weil. II

1980-12-01

Pierre Deligne

Real homotopy theory of K�hler manifolds

1975-10-01

Pierre Deligne , Phillip Griffiths , John Morgan +1 more

Representations of Reductive Groups Over Finite Fields

1976-01-01

Pierre Deligne , G. Lusztig

Les Schémas de Modules de Courbes Elliptiques

1973-01-01

Pierre Deligne , Michael Rapoport

Monodromy of hypergeometric functions and non-lattice integral monodromy

1986-12-01

Pierre Deligne , G. D. Mostow

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Hodge Cycles, Motives, and Shimura Varieties

1982-01-01

Pierre Deligne , J. S. Milne , Arthur Ogus +1 more

Formes modulaires et représentations e-adiques

2006-11-23

Pierre Deligne

Hodge Cycles, Motives, and Shimura Varieties

1982-01-01

Pierre Deligne , J. S. Milne , Arthur Ogus +1 more

Rel�vements modulop 2 et d�composition du complexe de de Rham

1987-06-01

Pierre Deligne , Luc Illusie

Cohomologie Etale

1977-01-01

Pierre Deligne

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Groupes fondamentaux motiviques de Tate mixte

2005-01-01

Pierre Deligne , A. B. Goncharov

Nous définissons la catégorie des motifs de Tate mixte sur l'anneau des S-entiers d'un corps de nombres, et le groupe fondamental motivique (rendu unipotent) d'une variété unirationnelle sur un corps … Nous définissons la catégorie des motifs de Tate mixte sur l'anneau des S-entiers d'un corps de nombres, et le groupe fondamental motivique (rendu unipotent) d'une variété unirationnelle sur un corps de nombres. Nous considérons plus en détail le groupe fondamental motivique de la droite projective moins 0, ∞ et les racines N-ièmes de l'unité. We define the category of mixed Tate motives over the ring of S-integers of a number field. We define the motivic fundamental group (made unipotent) of a unirational variety over a number field. We apply this to the study of the motivic fundamental group of the projective line punctured at zero, infinity and all Nth roots of unity.

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Values of abelianL-functions at negative integers over totally real fields

1980-10-01

Pierre Deligne , Kenneth A. Ribet

Les Constantes des Equations Fonctionnelles des Fonctions L

2009-02-27

Pierre Deligne

Travaux de Shimura

1971-01-01

Pierre Deligne

La conjecture de Weil pour les surfacesK3

1971-09-01

Pierre Deligne

Hodge Cycles on Abelian Varieties

1982-01-01

Pierre Deligne

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Valeurs de fonctions L et périodes d’intégrales

1979-01-01

Pierre Deligne

Valeurs de fonctions 𝐿 et périodes d’intégrales

1979-01-01

Pierre Deligne

Trace paley-wiener theorem for reductivep-adic groups

1986-12-01

Joseph Bernstein , Pierre Deligne , David Kazhdan

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Catégories tannakiennes

2007-01-01

Pierre Deligne

Dans [6], N. Saavedra décrit certaines catégories munies d'un produit tensoriel, les catégories tannakiennes (2.8), comme catégories de représentations de gerbes (cas particulier: représentations d'un schéma en groupes). Sa démonstration … Dans [6], N. Saavedra décrit certaines catégories munies d'un produit tensoriel, les catégories tannakiennes (2.8), comme catégories de représentations de gerbes (cas particulier: représentations d'un schéma en groupes). Sa démonstration est incomplète (cf. [2] 3.15). Notre but premier est de la compléter. Je n'ai pas su rédiger un exposé court ne donnant que les arguments manquants: bien des idées de l'article sont dans [6], dues à Saavedra et, par son intermédiaire, à A. Grothendieck.

Applications de la formule des traces aux sommes trigonométrigues

1977-01-01

Pierre Deligne

Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant

1994-01-01

Pierre Deligne , G. Pappas

© Foundation Compositio Mathematica, 1994, tous droits reserves. L’acces aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation … © Foundation Compositio Mathematica, 1994, tous droits reserves. L’acces aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

On the locus of Hodge classes

1995-01-01

Eduardo Cattani , Pierre Deligne , Aroldo Kaplan

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a nonsingular complex algebraic variety and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a nonsingular complex algebraic variety and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a polarized variation of Hodge structure of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 p"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with polarization form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Given an integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{S^{(K)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the space of pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis s comma u right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(s,u)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s element-of upper S"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">s \in S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u element-of script upper V Subscript s"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">u \in {\mathcal {V}_s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> integral of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p comma p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p,p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q left-parenthesis u comma u right-parenthesis less-than-or-equal-to upper K"> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Q(u,u) \leq K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show in Theorem 1.1 that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{S^{(K)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an algebraic variety, finite over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the local system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 2 p Baseline left-parenthesis upper X Subscript s Baseline comma double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^{2p}}({X_s},\mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>/torsion associated with a family of nonsingular projective varieties parametrized by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the result implies that the locus where a given integral class of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p comma p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p,p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> remains of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p comma p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p,p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is algebraic.

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Central extensions of reductive groups by K2

2001-12-01

Jean-Luc Brylinski , Pierre Deligne

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Action du groupe des tresses sur une cat�gorie

1997-03-19

Pierre Deligne

Déformations de l'Algèbre des Fonctions d'une Variété Symplectique: Comparaison entre Fedosov et De Wilde, Lecomte

1995-03-01

Pierre Deligne

Formes Modulaires et Representations De GL(2)

2009-02-27

Pierre Deligne

Commensurabilities among lattices in PU(1,n)

1993-01-01

Pierre Deligne , D. Mostow

The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n-variables. These are treated as an (n+1) dimensional vector … The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n-variables. These are treated as an (n+1) dimensional vector space of multivalued locally holomorphic functions defined on the space of n+3 tuples of distinct points on the projective line P modulo, the diagonal section of Auto P=m. For n=1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents three singular points.This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of PU(1, n).The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in PU(1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents the divisors at infinity permits one to prove generalizations in n-variables of the Kummer identities for n-1 involving quadratic and cubic changes of the variable.

Représentations des groupes réductifs sur un corps local

1984-01-01

Pierre Deligne , J.N. Bernstein , D. Kazhdan +1 more

Survey of Drinfel′d modules

1987-01-01

Pierre Deligne , Dale Husemöller

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Les constantes locales de l'�quation fonctionnelle de la fonctionL d'Artin d'une repr�sentation orthogonale

1976-12-01

Pierre Deligne

SGA 4 1/2 – Cohomologie étale

1977-01-01

Pierre Deligne

Relèvement des Surfaces K3 en Caractéristique Nulle

1981-01-01

Pierre Deligne , Luc Illusie

Le groupe fondamental du complément d'une courbe plane n'ayant que des points doubles ordinaires est abélien [d'après W. Fulton]

2006-11-24

Pierre Deligne

Un Théorème de Finitude Pour la Monodromie

1987-01-01

Pierre Deligne

Duality for representations of a reductive group over a finite field, II

1983-04-01

Pierre Deligne , G. Lusztig

Local behavior of Hodge structures at infinity

1996-09-26

Pierre Deligne , Brian Greene , Shing‐Tung Yau

Duality for representations of a reductive group over a finite field

1982-01-01

Pierre Deligne , G. Lusztig

Le groupe fondamental unipotent motivique de $\mathbb{G}_{m}-\mu_{N}$ , pour N=2,3,4,6 ou 8

2010-09-03

Pierre Deligne

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Formes modulaires et représentations ℓ-adiques

1969-01-01

Pierre Deligne

Commensurabilities among Lattices in PU (1,n). (AM-132)

1993-12-31

Pierre Deligne , G. D. Mostow

The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n -variables. These are treated as an ( n … The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n -variables. These are treated as an ( n +1) dimensional vector space of multivalued locally holomorphic functions defined on the space of n +3 tuples of distinct points on the projective line P modulo, the diagonal section of Auto P = m . For n =1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points. This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of PU (1, n ). The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in PU (1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations in n -variables of the Kummer identities for n -1 involving quadratic and cubic changes of the variable.

Counting local systems with principal unipotent local monodromy

2013-08-02

Pierre Deligne , Yuval Z. Flicker

Let X1 be a curve of genus g, projective and smooth over Fq.Let S1 ⊂ X1 be a reduced divisor consisting of N1 closed points of X1.Let (X, S) be … Let X1 be a curve of genus g, projective and smooth over Fq.Let S1 ⊂ X1 be a reduced divisor consisting of N1 closed points of X1.Let (X, S) be obtained from (X1, S1) by extension of scalars to an algebraic closure F of Fq.Fix a prime l not dividing q.The pullback by the Frobenius endomorphism Fr of X induces a permutation Fr * of the set of isomorphism classes of rank n irreducible Q l -local systems on X -S.It maps to itself the subset of those classes for which the local monodromy at each s ∈ S is unipotent, with a single Jordan block.Let T (X1, S1, n, m) be the number of fixed points of Fr * m acting on this subset.Under the assumption that N1 ≥ 2, we show that T (X1, S1, n, m) is given by a formula reminiscent of a Lefschetz fixed point formula: the function m → T (X1, S1, n, m) is of the form niγ m i for suitable integers ni and "eigenvalues" γi.We use Lafforgue to reduce the computation of T (X1, S1, n, m) to counting automorphic representations of GL(n), and the assumption N1 ≥ 2 to move the counting to the multiplicative group of a division algebra, where the trace formula is easier to use.

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Cohomologie étale: les points de départ

1977-01-01

Pierre Deligne , Jean-François Boutot

On the exceptional series, and its descendants

2002-11-12

Pierre Deligne , Benedict H. Gross

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Un fil d’Ariane pour SGA 4, SGA 4 1/2 et SGA 5

1977-01-01

Pierre Deligne

Motives, periods, and functoriality

2025-03-07

Pierre Deligne , A. Raghuram

Motives, Periods, and Functoriality

2024-03-05

Pierre Deligne , A. Raghuram

Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group respecting $\mathsf{s}$, we describe a functorial transfer … Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group respecting $\mathsf{s}$, we describe a functorial transfer $M^V$. We formulate a criterion that guarantees when the two periods of $M^V$ are equal. This has an implication for the critical values of the $L$-function attached to $M^V.$ The criterion is explicated in a variety of examples such as: tensor product motives and Rankin-Selberg $L$-functions; orthogonal motives and the standard $L$-function for even orthogonal groups; twisted tensor motives and Asai $L$-functions.

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Le critère d’Abel pour la résolubilité par radicaux d’une équation irréductible de degré premier

2021-09-17

Pierre Deligne

In his last letter to Crelle, Abel states a criterion for the solvability by radicals of an irreducible equation of prime degree. Sylow finds Abel’s statement ambiguous, and writes that … In his last letter to Crelle, Abel states a criterion for the solvability by radicals of an irreducible equation of prime degree. Sylow finds Abel’s statement ambiguous, and writes that it should be modified. We show the correctness of Abel’s original statement.

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On the -problem for restrictions of complex reflection arrangements

2020-01-20

Nils Amend , Pierre Deligne , Gerhard Röhrle

Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$ . It is known that the complement $X(\mathscr{A}(W))$ of … Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$ . It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$ , let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$ . We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$ . We prove it in case of the monomial groups $W=G(r,p,\ell )$ . Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

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Mathematical Autobiography

2019-01-01

Pierre Deligne

Comptage de faisceaux $l$-adiques

2018-11-06

Pierre Deligne

Extension des scalaires par le morphisme de Frobenius, pour les groupes réductifs

2018-05-24

Pierre Deligne

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Arithmetic monodromy actions on pro-metabelian fundamental groups of once-punctured elliptic curves

2017-10-16

William Yun Chen , Pierre Deligne

We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has … We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to congruence structures of level $H$. Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of braid-like outer with the subgroup of IA outer automorphisms which induce the identity on the abelianization. We also show a surprising result that all IA-automorphisms leave every open normal subgroup stable.

On complete reducibility in characteristic $p$

2017-09-01

V. Balaji , Pierre Deligne , A. J. Parameswaran

Let $G$ be a reductive group over a field $k$ which is algebraically closed of characteristic $p \neq 0$. We prove a structure theorem for a class of subgroup schemes … Let $G$ be a reductive group over a field $k$ which is algebraically closed of characteristic $p \neq 0$. We prove a structure theorem for a class of subgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. As applications, we derive semi-simplicity results, generalizing earlier results of Serre proven in 1998, and also obtain an analogue of Luna's \'etale slice theorem for suitable bounds on $p$.

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The first fundamental theorem of invariant theory for the orthosymplectic super group

2017-06-27

Pierre Deligne , G Lehrer , R. B. Zhang

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Arithmetic monodromy actions on pro-metabelian fundamental groups of once-punctured elliptic curves

2017-01-01

William Yun Chen , Pierre Deligne

We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has … We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "braid-like outer automorphisms" with the subgroup of "IA" outer automorphisms which induce the identity on the abelianization. We also show a surprising result that all IA-automorphisms leave every open normal subgroup stable.

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Semi-simplicité de produits tensoriels en caractéristique p

2013-11-11

Pierre Deligne

Counting local systems with principal unipotent local monodromy

2013-08-02

Pierre Deligne , Yuval Z. Flicker

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Le groupe fondamental unipotent motivique de $\mathbb{G}_{m}-\mu_{N}$ , pour N=2,3,4,6 ou 8

2010-09-03

Pierre Deligne

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Le théorème de plngement de Nagata

2010-01-01

Pierre Deligne

Le théorème de plongement de Nagata

2010-01-01

Pierre Deligne

Except for a few corrections added in 2010, the article reproduces notes, sent to Nagata in the 1970s, which were a translation in the language of schemes of his articles … Except for a few corrections added in 2010, the article reproduces notes, sent to Nagata in the 1970s, which were a translation in the language of schemes of his articles [2], [3]. The proofs are a “constructive” version of his proofs. Making the proofs constructives allowed us to proceed without using valuations. A more detailed exposition and additional results are in Conrad [1]. Unless otherwise mentioned, all the schemes considered are supposed to be Noetherian.

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Motivic Eilenberg-MacLane spaces . Le groupe fondamental unipotent motivique de G[m] - μ[N], pour N=2, 3, 4, 6, ou 8 . Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations . A geometric criterion for generating the Fukaya category

2010-01-01

Vladimir Voevodsky , Pierre Deligne , Anton Alekseev +3 more

Formes Modulaires et Representations De GL(2)

2009-02-27

Pierre Deligne

Les Constantes des Equations Fonctionnelles des Fonctions L

2009-02-27

Pierre Deligne

Voevodsky's Lectures on Motivic Cohomology 2000/2001

2009-01-01

Pierre Deligne

Equivalence numerique, equivalence cohomologique, et theorie de Lefschetz des varietes abeliennes sur les corps finis

2009-01-01

Laurent Clozel , Pierre Deligne

We determine the group of automorphisms generated, in the endomorphisms of the cohomology of an Abelian variety on a finite field, by the Lefschetz operators and the complex multiplications. We determine the group of automorphisms generated, in the endomorphisms of the cohomology of an Abelian variety on a finite field, by the Lefschetz operators and the complex multiplications.

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Extended Euler congruence

2008-11-28

Pierre Deligne

Catégories tannakiennes

2007-01-01

Pierre Deligne

Les difféomorphismes du cercle [d’après M. R. Herman]

2006-11-24

Pierre Deligne

Le groupe fondamental du complément d'une courbe plane n'ayant que des points doubles ordinaires est abélien [d'après W. Fulton]

2006-11-24

Pierre Deligne

Formes modulaires et représentations e-adiques

2006-11-23

Pierre Deligne

Travaux de Griffiths

2006-11-23

Pierre Deligne

Les constantes des equations fonctionnelles des fonctions L

2006-11-23

Pierre Deligne

Groupes fondamentaux motiviques de Tate mixte

2005-01-01

Pierre Deligne , A. B. Goncharov

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Appendix to ``Deligne's integrality theorem in unequal characteristic and rational points over finite fields'' by Hélène Esnault

2004-01-01

Pierre Deligne , Hélène Esnault

One extends P. Deligne's notion of integrality over a finite field for a $\ell$-adic sheaf on a scheme of finite type over a local field with finite residue field. One … One extends P. Deligne's notion of integrality over a finite field for a $\ell$-adic sheaf on a scheme of finite type over a local field with finite residue field. One shows that this integrality notion is preserved by $Rf_!$, as it is over a finite field after P. Deligne.

Groupes fondamentaux motiviques de Tate mixte

2003-01-01

Pierre Deligne , A. B. Goncharov

We define the category of mixed Tate motives over the ring of S-integers of a number field. We define the motivic fundamental group (made unipotent) of a unirational variety over … We define the category of mixed Tate motives over the ring of S-integers of a number field. We define the motivic fundamental group (made unipotent) of a unirational variety over a number field. We apply this to the study of the motivic fundamental group of the projective line punctured at zero, infinity and all N-th roots of unity.

On the exceptional series, and its descendants

2002-11-12

Pierre Deligne , Benedict H. Gross

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Central extensions of reductive groups by K2

2001-12-01

Jean-Luc Brylinski , Pierre Deligne

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Central extensions of reductive groups by K[2] . Spectral curves, opers and integrable systems . Variétés sphériques de type A

2001-01-01

Jean-Luc Brylinski , Pierre Deligne , Dani Ben‐Zvi +2 more

L'algèbre de cohomologie du complément, dans un espace affine, d'une famille finie de sous-espaces affines.

2000-01-01

Pierre Deligne , Mark Goresky , Robert MacPherson

L'algèbre de L'algèbre de

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Supersolutions

1999-01-01

Pierre Deligne , Daniel S. Freed

We develop classical globally supersymmetric theories. As much as possible, we treat various dimensions and various amounts of supersymmetry in a uniform manner. We discuss theories both in components and … We develop classical globally supersymmetric theories. As much as possible, we treat various dimensions and various amounts of supersymmetry in a uniform manner. We discuss theories both in components and in superspace. Throughout we emphasize geometric aspects. The beginning chapters give a general discussion about supersymmetric field theories; then we move on to detailed computations of lagrangians, etc. in specific theories. An appendix details our sign conventions. This text will appear in a two-volume work "Quantum Fields and Strings: A Course for Mathematicians" to be published soon by the American Mathematical Society. Some of the cross-references may be found at http://www.math.ias.edu/~drm/QFT/

Action du groupe des tresses sur une cat�gorie

1997-03-19

Pierre Deligne

Extensions centrales de groupes algébriques simplement connexes et cohomologie galoisienne

1996-12-01

Pierre Deligne

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Local behavior of Hodge structures at infinity

1996-09-26

Pierre Deligne , Brian Greene , Shing‐Tung Yau

On the Locus of Hodge Classes

1995-04-01

Eduardo Cattani , Pierre Deligne , Aroldo Kaplan

Let $S$ be a nonsingular complex algebraic variety and $\mathcal {V}$ a polarized variation of Hodge structure of weight $2p$ with polarization form $Q$. Given an integer $K$, let ${S^{(K)}}$ … Let $S$ be a nonsingular complex algebraic variety and $\mathcal {V}$ a polarized variation of Hodge structure of weight $2p$ with polarization form $Q$. Given an integer $K$, let ${S^{(K)}}$ be the space of pairs $(s,u)$ with $s \in S$, $u \in {\mathcal {V}_s}$ integral of type $(p,p)$, and $Q(u,u) \leq K$. We show in Theorem 1.1 that ${S^{(K)}}$ is an algebraic variety, finite over $S$. When $\mathcal {V}$ is the local system ${H^{2p}}({X_s},\mathbb {Z})$/torsion associated with a family of nonsingular projective varieties parametrized by $S$, the result implies that the locus where a given integral class of type $(p,p)$ remains of type $(p,p)$ is algebraic.

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Déformations de l'Algèbre des Fonctions d'une Variété Symplectique: Comparaison entre Fedosov et De Wilde, Lecomte

1995-03-01

Pierre Deligne

On the locus of Hodge classes

1995-01-01

Eduardo Cattani , Pierre Deligne , Aroldo Kaplan

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On the Locus of Hodge Classes

1994-01-01

Eduardo Cattani , Pierre Deligne , Aroldo Kaplan

Let $f: X \rightarrow S$ be a family of non singular projective varieties parametrized by a complex algebraic variety $S$. Fix $s \in S$, an integer $p$, and a class … Let $f: X \rightarrow S$ be a family of non singular projective varieties parametrized by a complex algebraic variety $S$. Fix $s \in S$, an integer $p$, and a class $h \in {\rm H}^{2p}(X_s,\Z)$ of Hodge type $(p,p)$. We show that the locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This result, which in the geometric case would follow from the rational Hodge conjecture, is obtained in the setting of variations of Hodge structures.

Structures de Hodge mixtes réelles

1994-01-01

Pierre Deligne

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Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant

1994-01-01

Pierre Deligne , G. Pappas

§8. PRELIMINARIES ON MONODROMY GROUPS

1993-12-31

Pierre Deligne , G. D. Mostow

§5. GELFAND’S DESCRIPTION OF HYPERGEOMETRIC FUNCTIONS

1993-12-31

Pierre Deligne , G. D. Mostow

§16. LIVNE’S CONSTRUCTION OF LATTICES IN PU(1,2)

1993-12-31

Pierre Deligne , G. D. Mostow

§4. LAURICELLA’S HYPERGEOMETRIC FUNCTIONS

1993-12-31

Pierre Deligne , G. D. Mostow

Commensurabilities among Lattices in PU (1,n). (AM-132)

1993-12-31

Pierre Deligne , G. D. Mostow

The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n -variables. These are treated as an ( n … The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n -variables. These are treated as an ( n +1) dimensional vector space of multivalued locally holomorphic functions defined on the space of n +3 tuples of distinct points on the projective line P modulo, the diagonal section of Auto P = m . For n =1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points. This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of PU (1, n ). The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in PU (1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations in n -variables of the Kummer identities for n -1 involving quadratic and cubic changes of the variable.

Coauthor	Papers Together
G. D. Mostow	7
J. S. Milne	4
Kuang-yen Shih	4
Arthur Ogus	4
Eduardo Cattani	3
Aroldo Kaplan	3
G. Lusztig	3
Jean-Luc Brylinski	2
Laurent Clozel	2
Luc Illusie	2
A. B. Goncharov	2
William Yun Chen	2
Michael Rapoport	1
Kenneth A. Ribet	1
Guy Henniart	1
John N. Mather	1
A. Raghuram	1
Jean-Claude Tougeron	1
A. J. Parameswaran	1
J. S. Milne	1
Mark Goresky	1
Anthony N. Godwin	1
N. Iwahori	1
Yuval Z. Flicker	1
Gerhard Röhrle	1
Vladimir Voevodsky	1
和也加藤	1
Moss E. Sweedler	1
D. Kazhdan	1
Dani Ben‐Zvi	1
John Morgan	1
G. Pappas	1
Brian Greene	1
Benedict H. Gross	1
Hélène Esnault	1
Mohammed Abouzaid	1
Robert W. Kilmoyer	1
Harold Widom	1
G Lehrer	1
Daniel S. Freed	1
J.N. Bernstein	1
A. Raghuram	1
Anton Alekseev	1
D. Luna	1
Ronald G. Douglas	1
James Arthur	1
Benjamin Enriquez	1
Phillip Griffiths	1
Charles W. Curtis	1
Spencer Bloch	1

Théorie de hodge, III

1974-01-01

Pierre Deligne

Théorème de Lefschetz et Critères de Dégénérescence de Suites Spectrales

1968-12-01

P. Deligne

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Nilpotent connections and the monodromy theorem: Applications of a result of turrittin

1970-01-01

Nicholas M. Katz

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Equations Différentielles à Points Singuliers Réguliers

1970-01-01

Pierre Deligne

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Geometric Invariant Theory

1965-01-01

David Mumford

Théorie de Hodge, II

1971-12-01

Pierre Deligne

Variétés abéliennes et courbes algébriques

1948-01-01

André Weil

Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems

1970-01-01

Phillip Griffiths

Part I. Summary of main results 231 1.The geometric situation giving rise to variation of Hodge structure.... 231 2. Data given by the variation of Hodge structure 232 3. Theorems … Part I. Summary of main results 231 1.The geometric situation giving rise to variation of Hodge structure.... 231 2. Data given by the variation of Hodge structure 232 3. Theorems about monodromy of homology 235 4. Theorems about Picard-Fuchs equations (Gauss-Manin connexion).... 237 5. Global theorems about holomorphic and locally constant cohomology classes 242 6. Global results on variation of Hodge structure 246 Part II.Problems and conjectures 247 7. Problems on Torelli-type theorems 247 8. Problems on local monodromy and variation of Hodge structure 248 9. Questions on compactification and the behavior of periods at infinity. .

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Valeurs de fonctions 𝐿 et périodes d’intégrales

1979-01-01

Pierre Deligne

On the Periods of Certain Rational Integrals: I

1969-11-01

Philip A. Griffiths

In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs … In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how Macaulay's theorem 4.11 is essentially equivalent to a suitable vanishing theorem for sheaf cohomology. We shall also give a proof of the de Rham algebraic theorem used in the proof of Theorem 5.3. However, our principal motivation is to be able to discuss rational integrals in case our hypersurface V c P, has rather simple singularities, and the localization technique of sheaf theory seems to be the best method for doing this (cf. ?? 15, 16 below). Let then V c P, be a non-singular hypersurface. We want to give a sheaf-theoretic version of the Hodge filtration of Hq(V, C) (cf. ? 8). For this we let Qv be the sheaf on V of holomorphic q-forms and & c QV the subsheaf of closed forms. The Poincare' lemma for holomorphic differentials gives an exact sequence

Geometric Invariant Theory

1994-01-01

David Mumford , John E. Fogarty , Frances Kirwan

La conjecture de Weil. I

1974-12-01

Pierre Deligne

On the differentiation of De Rham cohomology classes with respect to parameters

1968-01-01

Nicholas M. Katz , Tadao Oda

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Abelian l-Adic Representations and Elliptic Curves

1997-11-15

Jean-Pierre Serre

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on … This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Good Reduction of Abelian Varieties

1968-11-01

Jean-Pierre Serre , John Tate

La conjecture de Weil pour les surfacesK3

1971-09-01

Pierre Deligne

Géométrie algébrique et géométrie analytique

1956-01-01

Jean-Pierre Serre

Toute variété algébrique X sur le corps des nombres complexes peut être munie, de façon canonique, d'une structure d'espace analytique ; tout faisceau algébrique cohérent sur X détermine un faisceau … Toute variété algébrique X sur le corps des nombres complexes peut être munie, de façon canonique, d'une structure d'espace analytique ; tout faisceau algébrique cohérent sur X détermine un faisceau analytique cohérent. Lorsque X est une variété projective, nous montrons que, réciproquement, tout faisceau analytique cohérent sur X peut être obtenu ainsi, et de façon unique ; de plus, cette correspondance préserve les groupes de cohomologie. Ces résultats contiennent comme cas particuliers des théorèmes classiques de Chow et Lefschetz, et permettent d'aborder la comparaison entre espaces fibrés algébriques et espaces fibrés analytiques de base une variété algébrique projective.

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On the de rham cohomology of algebraic varieties

1966-01-01

A. Grothendieck

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Les Constantes des Equations Fonctionnelles des Fonctions L

2009-02-27

Pierre Deligne

Automorphic Forms on GL (2)

1970-01-01

H. Jacquet , R. P. Langlands

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Abelian varieties attached to polarized K3-surfaces

1967-12-01

Ichirō Satake

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The algebraic topology of smooth algebraic varieties

1978-12-01

John W. Morgan

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Cohomologie non abélienne

1971-01-01

Jean Giraud

Variation of hodge structure: The singularities of the period mapping

1973-09-01

Wilfried Schmid

Théorie des Topos et Cohomologie Etale des Schémas

1972-01-01

Alexandre Grothendieck , J. L. Verdier

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Introduction to Affine Group Schemes

1979-01-01

William C. Waterhouse

Compactification of Arithmetic Quotients of Bounded Symmetric Domains

1966-11-01

Walter L. Baily , Armand Borel

Algebraic approximation of structures over complete local rings

1969-01-01

Michael Artin

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Cohomologie Etale

1977-01-01

Pierre Deligne

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Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

1970-01-01

Phillip Griffiths

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Théorie des Topos et Cohomologie Etale des Schémas

1972-01-01

J. L. Verdier , B. Saint-Donat , Alexandre Grothendieck +1 more

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Categories for the Working Mathematician

1971-01-01

Saunders Mac Lane

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Complexe de de\thinspace Rham-Witt et cohomologie cristalline

1979-01-01

Luc Illusie

chier doit contenir la présente mention de copyright. chier doit contenir la présente mention de copyright.

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The irreducibility of the space of curves of given genus

1969-01-01

Pierre Deligne , D. Mumford

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Doubles mélanges des polylogarithmes multiples aux racines de l’unité

2002-07-01

Georges Racinet

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Functors of Artin rings

1968-01-01

Michael Schlessinger

Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation.Let A be a complete noetherian local ring, u, … Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation.Let A be a complete noetherian local ring, u, its maximal ideal, and k = A/u.the residue field.(In most applications A is k itself, or a ring of Witt vectors.)Let C be the category of Artin local A-algebras with residue field k.A covariant functor F from C to Sets is called pro-representable if it has the formwhere R is a complete local A-algebra such that R/mn is in C, all n.(m is the maximal ideal in R.)In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, ■)->■ F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism.Theorem 2.11, §2, gives a criterion for F to have a hull, and also a simple criterion for pro-representability which avoids the use of Grothendieck's techniques of nonflat descent [3], in some cases.Grothendieck's program is carried out by Levelt in [4].§3 contains a few geometric applications of these results.To avoid awkward terminology, I have used the word " pro-representable " in a more restrictive sense than Grothendieck [3] has.He considers the category of A-algebras of finite length and allows R to be a projective limit of such rings.The methods of this paper are a simple extension of those used by David Mumford in a proof (unpublished) of the existence of formal moduli for polarized Abelian varieties.I am indebted to Mumford and to John Täte for many valuable suggestions.1.The category CA.Let A be a complete noetherian local ring, with maximal ideal u. and residue field k = A/u.. We define C= CA to be the category of Artinian local A-algebras having residue field k. (That is, the "structure morphism" A -> ,4 of such a ring A induces a trivial extension of residue fields.)Morphisms in C are local homomorphisms of A-algebras.

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The duality operation in the character ring of a finite Chevalley group

1979-01-01

Dean Alvis

It is possible (as in [4] ) to define a duality operation f -• f * in the ring of virtual characters of an arbitrary finite group with a split … It is possible (as in [4] ) to define a duality operation f -• f * in the ring of virtual characters of an arbitrary finite group with a split (B, 7V)-pair of characteristic p.Such a group arises as the fixed points under a Frobenius map of a connected reductive algebraic group, defined over a finite field [1].This paper contains statements of several general properties of the duality map f -• f * and two related operations (see § §2 and 4).The duality map f -• f * generalizes the construction in [2] of the Steinberg character, and interacts well with the organization of the characters from the point of view of cuspidal characters ( §6).It is hoped that there is also a useful interaction with the Deligne-Lusztig virtual characters R^O.Partial results have been obtained in this direction ( §5).Detailed proofs will appear elsewhere.

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Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem

1972-01-01

Armand Borel

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The dihedral Lie algebras and Galois symmetries of $\pi_1^{(l)}(\mathbb{P}-(\{0,\infty\}\cup \mu_N))$

2001-12-01

A. B. Goncharov

We describe the image of the absolute Galois group acting on the pro-$l$ completion of the fundamental group of the $\mathbb {G}_m$ minus $N$th roots of unity. We relate the … We describe the image of the absolute Galois group acting on the pro-$l$ completion of the fundamental group of the $\mathbb {G}_m$ minus $N$th roots of unity. We relate the structure of the image with geometry and topology of modular varieties for the congruence subgroups $\Gamma_1(m;N)$ of ${\rm GL}_m(\mathbb {Z})$ for $m=1,2,3\ldots$.

Rational singularities, with applications to algebraic surfaces and unique factorization

1969-01-01

Joseph Lipman

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The invariant theory of n × n matrices

1976-03-01

Claudio Procesi

Formule de Lefschetz et rationalité des fonctions $L$

1966-01-01

Alexander Grothendieck